MinT - Best Known (205−19, 205, s)-Nets in Base 3

Best Known (205−19, 205, s)-Nets in Base 3

(205−19, 205, 932122)-Net over F₃ — Constructive and digital

Digital (186, 205, 932122)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 24, 56)-net over F₃, using
  - trace code for nets [i] based on digital (3, 12, 28)-net over F₉, using
    - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
        the Hermitian function field over F₉ [i]
2. digital (162, 181, 932066)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸¹, 932066, F₃, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(3¹⁸¹, 8388595, F₃, 19) (dual of [8388595, 8388414, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(205−19, 205, 3845767)-Net over F₃ — Digital

Digital (186, 205, 3845767)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁵, 3845767, F₃, 2, 19) (dual of [(3845767, 2), 7691329, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁰⁵, 4194357, F₃, 2, 19) (dual of [(4194357, 2), 8388509, 20]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3²⁴, 56, F₃, 2, 9) (dual of [(56, 2), 88, 10]-NRT-code), using
      - extracting embedded OOA [i] based on digital (15, 24, 56)-net over F₃, using
        trace code for nets [i] based on digital (3, 12, 28)-net over F₉, using
        net from sequence [i] based on digital (3, 27)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
        the Hermitian function field over F₉ [i]
    2. linear OOA(3¹⁸¹, 4194301, F₃, 2, 19) (dual of [(4194301, 2), 8388421, 20]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3¹⁸¹, 8388602, F₃, 19) (dual of [8388602, 8388421, 20]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(205−19, 205, large)-Net in Base 3 — Upper bound on s

There is no (186, 205, large)-net in base 3, because

17 times m-reduction [i] would yield (186, 188, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]